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In part I of this paper we studied additive decomposability of the set $\F_y$of th $y$-smooth numbers and the multiplicative decomposability of the shiftedset $\g_y=\F_y+\{1\}$. In this paper, focusing on the case of 'large' functions$y$, we continue the study of these problems. Further, we also investigate aproblem related to the m-decomposability of $k$-term sumsets, for arbitrary$k$.

On additive and multiplicative decompositions of sets of integers with restricted prime factors, II. (Smooth numbers and generalizations.)